Abstract. We investigate the connection between metastability and the
spectrum near zero corresponding to the elliptic, second-order,
differential operator
$L_\e\equiv -\e\D+\nabla F\cdot\nabla$, $\e>0$,
with $F:\R^d\to\R$. For generic $F$ to each local minimum of $F$
there corresponds an eigenvalue of $L_\e$ which converges to
zero exponentially fast as $\e\downarrow 0$. Modulo errors
of exponentially small order in $\e$ this eigenvalue is
given as the inverse of the expected metastable relaxation
time. The corresponding eigenstate, which may be
viewed as a metastable state, is highly concentrated in the
basin of attraction of this trap.